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      SUBROUTINE <a name="SLARRB.1"></a><a href="slarrb.f.html#SLARRB.1">SLARRB</a>( N, D, LLD, IFIRST, ILAST, RTOL1,
     $                   RTOL2, OFFSET, W, WGAP, WERR, WORK, IWORK,
     $                   PIVMIN, SPDIAM, TWIST, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            IFIRST, ILAST, INFO, N, OFFSET, TWIST
      REAL               PIVMIN, RTOL1, RTOL2, SPDIAM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      INTEGER            IWORK( * )
      REAL               D( * ), LLD( * ), W( * ),
     $                   WERR( * ), WGAP( * ), WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Given the relatively robust representation(RRR) L D L^T, <a name="SLARRB.22"></a><a href="slarrb.f.html#SLARRB.1">SLARRB</a>
</span><span class="comment">*</span><span class="comment">  does &quot;limited&quot; bisection to refine the eigenvalues of L D L^T,
</span><span class="comment">*</span><span class="comment">  W( IFIRST-OFFSET ) through W( ILAST-OFFSET ), to more accuracy. Initial
</span><span class="comment">*</span><span class="comment">  guesses for these eigenvalues are input in W, the corresponding estimate
</span><span class="comment">*</span><span class="comment">  of the error in these guesses and their gaps are input in WERR
</span><span class="comment">*</span><span class="comment">  and WGAP, respectively. During bisection, intervals
</span><span class="comment">*</span><span class="comment">  [left, right] are maintained by storing their mid-points and
</span><span class="comment">*</span><span class="comment">  semi-widths in the arrays W and WERR respectively.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D       (input) REAL             array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The N diagonal elements of the diagonal matrix D.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LLD     (input) REAL             array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          The (N-1) elements L(i)*L(i)*D(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IFIRST  (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The index of the first eigenvalue to be computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ILAST   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The index of the last eigenvalue to be computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RTOL1   (input) REAL            
</span><span class="comment">*</span><span class="comment">  RTOL2   (input) REAL            
</span><span class="comment">*</span><span class="comment">          Tolerance for the convergence of the bisection intervals.
</span><span class="comment">*</span><span class="comment">          An interval [LEFT,RIGHT] has converged if
</span><span class="comment">*</span><span class="comment">          RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
</span><span class="comment">*</span><span class="comment">          where GAP is the (estimated) distance to the nearest
</span><span class="comment">*</span><span class="comment">          eigenvalue.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  OFFSET  (input) INTEGER
</span><span class="comment">*</span><span class="comment">          Offset for the arrays W, WGAP and WERR, i.e., the IFIRST-OFFSET
</span><span class="comment">*</span><span class="comment">          through ILAST-OFFSET elements of these arrays are to be used.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  W       (input/output) REAL             array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On input, W( IFIRST-OFFSET ) through W( ILAST-OFFSET ) are
</span><span class="comment">*</span><span class="comment">          estimates of the eigenvalues of L D L^T indexed IFIRST throug
</span><span class="comment">*</span><span class="comment">          ILAST.
</span><span class="comment">*</span><span class="comment">          On output, these estimates are refined.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WGAP    (input/output) REAL             array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          On input, the (estimated) gaps between consecutive
</span><span class="comment">*</span><span class="comment">          eigenvalues of L D L^T, i.e., WGAP(I-OFFSET) is the gap between
</span><span class="comment">*</span><span class="comment">          eigenvalues I and I+1. Note that if IFIRST.EQ.ILAST
</span><span class="comment">*</span><span class="comment">          then WGAP(IFIRST-OFFSET) must be set to ZERO.
</span><span class="comment">*</span><span class="comment">          On output, these gaps are refined.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WERR    (input/output) REAL             array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On input, WERR( IFIRST-OFFSET ) through WERR( ILAST-OFFSET ) are
</span><span class="comment">*</span><span class="comment">          the errors in the estimates of the corresponding elements in W.
</span><span class="comment">*</span><span class="comment">          On output, these errors are refined.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) REAL             array, dimension (2*N)
</span><span class="comment">*</span><span class="comment">          Workspace.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IWORK   (workspace) INTEGER array, dimension (2*N)
</span><span class="comment">*</span><span class="comment">          Workspace.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  PIVMIN  (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The minimum pivot in the Sturm sequence.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SPDIAM  (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The spectral diameter of the matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TWIST   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The twist index for the twisted factorization that is used
</span><span class="comment">*</span><span class="comment">          for the negcount.
</span><span class="comment">*</span><span class="comment">          TWIST = N: Compute negcount from L D L^T - LAMBDA I = L+ D+ L+^T
</span><span class="comment">*</span><span class="comment">          TWIST = 1: Compute negcount from L D L^T - LAMBDA I = U- D- U-^T
</span><span class="comment">*</span><span class="comment">          TWIST = R: Compute negcount from L D L^T - LAMBDA I = N(r) D(r) N(r)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          Error flag.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Based on contributions by
</span><span class="comment">*</span><span class="comment">     Beresford Parlett, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">     Jim Demmel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">     Inderjit Dhillon, University of Texas, Austin, USA
</span><span class="comment">*</span><span class="comment">     Osni Marques, LBNL/NERSC, USA
</span><span class="comment">*</span><span class="comment">     Christof Voemel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      REAL               ZERO, TWO, HALF
      PARAMETER        ( ZERO = 0.0E0, TWO = 2.0E0,
     $                   HALF = 0.5E0 )
      INTEGER   MAXITR
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I, I1, II, IP, ITER, K, NEGCNT, NEXT, NINT,
     $                   OLNINT, PREV, R
      REAL               BACK, CVRGD, GAP, LEFT, LGAP, MID, MNWDTH,
     $                   RGAP, RIGHT, TMP, WIDTH
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      INTEGER            <a name="SLANEG.126"></a><a href="slaneg.f.html#SLANEG.1">SLANEG</a>
      EXTERNAL           <a name="SLANEG.127"></a><a href="slaneg.f.html#SLANEG.1">SLANEG</a>
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, MAX, MIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
<span class="comment">*</span><span class="comment">
</span>      MAXITR = INT( ( LOG( SPDIAM+PIVMIN )-LOG( PIVMIN ) ) /
     $           LOG( TWO ) ) + 2
      MNWDTH = TWO * PIVMIN
<span class="comment">*</span><span class="comment">
</span>      R = TWIST
      IF((R.LT.1).OR.(R.GT.N)) R = N
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Initialize unconverged intervals in [ WORK(2*I-1), WORK(2*I) ].
</span><span class="comment">*</span><span class="comment">     The Sturm Count, Count( WORK(2*I-1) ) is arranged to be I-1, while
</span><span class="comment">*</span><span class="comment">     Count( WORK(2*I) ) is stored in IWORK( 2*I ). The integer IWORK( 2*I-1 )
</span><span class="comment">*</span><span class="comment">     for an unconverged interval is set to the index of the next unconverged
</span><span class="comment">*</span><span class="comment">     interval, and is -1 or 0 for a converged interval. Thus a linked
</span><span class="comment">*</span><span class="comment">     list of unconverged intervals is set up.
</span><span class="comment">*</span><span class="comment">
</span>      I1 = IFIRST
<span class="comment">*</span><span class="comment">     The number of unconverged intervals
</span>      NINT = 0
<span class="comment">*</span><span class="comment">     The last unconverged interval found
</span>      PREV = 0

      RGAP = WGAP( I1-OFFSET )
      DO 75 I = I1, ILAST
         K = 2*I
         II = I - OFFSET
         LEFT = W( II ) - WERR( II )
         RIGHT = W( II ) + WERR( II )
         LGAP = RGAP
         RGAP = WGAP( II )
         GAP = MIN( LGAP, RGAP )

<span class="comment">*</span><span class="comment">        Make sure that [LEFT,RIGHT] contains the desired eigenvalue
</span><span class="comment">*</span><span class="comment">        Compute negcount from dstqds facto L+D+L+^T = L D L^T - LEFT
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Do while( NEGCNT(LEFT).GT.I-1 )
</span><span class="comment">*</span><span class="comment">
</span>         BACK = WERR( II )
 20      CONTINUE
         NEGCNT = <a name="SLANEG.174"></a><a href="slaneg.f.html#SLANEG.1">SLANEG</a>( N, D, LLD, LEFT, PIVMIN, R )
         IF( NEGCNT.GT.I-1 ) THEN
            LEFT = LEFT - BACK
            BACK = TWO*BACK
            GO TO 20
         END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Do while( NEGCNT(RIGHT).LT.I )
</span><span class="comment">*</span><span class="comment">        Compute negcount from dstqds facto L+D+L+^T = L D L^T - RIGHT
</span><span class="comment">*</span><span class="comment">
</span>         BACK = WERR( II )
 50      CONTINUE

         NEGCNT = <a name="SLANEG.187"></a><a href="slaneg.f.html#SLANEG.1">SLANEG</a>( N, D, LLD, RIGHT, PIVMIN, R )
          IF( NEGCNT.LT.I ) THEN
             RIGHT = RIGHT + BACK
             BACK = TWO*BACK
             GO TO 50
          END IF
         WIDTH = HALF*ABS( LEFT - RIGHT )
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( WIDTH.LE.CVRGD .OR. WIDTH.LE.MNWDTH ) THEN
<span class="comment">*</span><span class="comment">           This interval has already converged and does not need refinement.
</span><span class="comment">*</span><span class="comment">           (Note that the gaps might change through refining the
</span><span class="comment">*</span><span class="comment">            eigenvalues, however, they can only get bigger.)
</span><span class="comment">*</span><span class="comment">           Remove it from the list.
</span>            IWORK( K-1 ) = -1
<span class="comment">*</span><span class="comment">           Make sure that I1 always points to the first unconverged interval
</span>            IF((I.EQ.I1).AND.(I.LT.ILAST)) I1 = I + 1
            IF((PREV.GE.I1).AND.(I.LE.ILAST)) IWORK( 2*PREV-1 ) = I + 1
         ELSE
<span class="comment">*</span><span class="comment">           unconverged interval found
</span>            PREV = I
            NINT = NINT + 1
            IWORK( K-1 ) = I + 1
            IWORK( K ) = NEGCNT
         END IF
         WORK( K-1 ) = LEFT
         WORK( K ) = RIGHT
 75   CONTINUE

<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Do while( NINT.GT.0 ), i.e. there are still unconverged intervals
</span><span class="comment">*</span><span class="comment">     and while (ITER.LT.MAXITR)
</span><span class="comment">*</span><span class="comment">
</span>      ITER = 0
 80   CONTINUE
      PREV = I1 - 1
      I = I1
      OLNINT = NINT

      DO 100 IP = 1, OLNINT
         K = 2*I
         II = I - OFFSET
         RGAP = WGAP( II )
         LGAP = RGAP
         IF(II.GT.1) LGAP = WGAP( II-1 )
         GAP = MIN( LGAP, RGAP )
         NEXT = IWORK( K-1 )
         LEFT = WORK( K-1 )
         RIGHT = WORK( K )
         MID = HALF*( LEFT + RIGHT )

<span class="comment">*</span><span class="comment">        semiwidth of interval
</span>         WIDTH = RIGHT - MID
         TMP = MAX( ABS( LEFT ), ABS( RIGHT ) )
         CVRGD = MAX(RTOL1*GAP,RTOL2*TMP)
         IF( ( WIDTH.LE.CVRGD ) .OR. ( WIDTH.LE.MNWDTH ).OR.
     $       ( ITER.EQ.MAXITR ) )THEN
<span class="comment">*</span><span class="comment">           reduce number of unconverged intervals
</span>            NINT = NINT - 1
<span class="comment">*</span><span class="comment">           Mark interval as converged.
</span>            IWORK( K-1 ) = 0
            IF( I1.EQ.I ) THEN
               I1 = NEXT
            ELSE
<span class="comment">*</span><span class="comment">              Prev holds the last unconverged interval previously examined
</span>               IF(PREV.GE.I1) IWORK( 2*PREV-1 ) = NEXT
            END IF
            I = NEXT
            GO TO 100
         END IF
         PREV = I
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Perform one bisection step
</span><span class="comment">*</span><span class="comment">
</span>         NEGCNT = <a name="SLANEG.261"></a><a href="slaneg.f.html#SLANEG.1">SLANEG</a>( N, D, LLD, MID, PIVMIN, R )
         IF( NEGCNT.LE.I-1 ) THEN
            WORK( K-1 ) = MID
         ELSE
            WORK( K ) = MID
         END IF
         I = NEXT
 100  CONTINUE
      ITER = ITER + 1
<span class="comment">*</span><span class="comment">     do another loop if there are still unconverged intervals
</span><span class="comment">*</span><span class="comment">     However, in the last iteration, all intervals are accepted
</span><span class="comment">*</span><span class="comment">     since this is the best we can do.
</span>      IF( ( NINT.GT.0 ).AND.(ITER.LE.MAXITR) ) GO TO 80
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     At this point, all the intervals have converged
</span>      DO 110 I = IFIRST, ILAST
         K = 2*I
         II = I - OFFSET
<span class="comment">*</span><span class="comment">        All intervals marked by '0' have been refined.
</span>         IF( IWORK( K-1 ).EQ.0 ) THEN
            W( II ) = HALF*( WORK( K-1 )+WORK( K ) )
            WERR( II ) = WORK( K ) - W( II )
         END IF
 110  CONTINUE
<span class="comment">*</span><span class="comment">
</span>      DO 111 I = IFIRST+1, ILAST
         K = 2*I
         II = I - OFFSET
         WGAP( II-1 ) = MAX( ZERO,
     $                     W(II) - WERR (II) - W( II-1 ) - WERR( II-1 ))
 111  CONTINUE

      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="SLARRB.296"></a><a href="slarrb.f.html#SLARRB.1">SLARRB</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

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